Question

If p-1 is a factor of p^4-p^2-p-k, the value of k is

Answer 1

Use the remainder theorem: it says that if x - c is a linear factor of a polynomial f(x), then the remainder upon dividing f(x) by x - c is equal to f(c).

In this case,

f(p) = p⁴ - p² - p - k

and p - 1 is a factor of f(p). We have

f (1) = 1⁴ - 1² - 1 - k = -(k + 1)

Because p - 1 divides f(p), the remainder is 0, which means

-(k + 1) = 0

k + 1 = 0

k = -1